In physics, acoustics, and telecommunications, a harmonic is a sinusoidal wave with a frequency that is a positive integer multiple of the fundamental frequency of a periodic signal. The fundamental frequency is also called the 1st harmonic; the other harmonics are known as higher harmonics. As all harmonics are periodic at the fundamental frequency, the sum of harmonics is also periodic at that frequency. The set of harmonics forms a harmonic series. The term is employed in various disciplines, including music, physics, acoustics, electronic power transmission, radio technology, and other fields. For example, if the fundamental frequency is 50 Hz, a common AC power supply frequency, the frequencies of the first three higher harmonics are 100 Hz (2nd harmonic), 150 Hz (3rd harmonic), 200 Hz (4th harmonic) and any addition of waves with these frequencies is periodic at 50 Hz. An n {\displaystyle \ n} th characteristic mode, for n > 1 , {\displaystyle \ n>1\ ,} will have nodes that are not vibrating. For example, the 3rd characteristic mode will have nodes at 1 3 L {\displaystyle \ {\tfrac {1}{3}}\ L\ } and 2 3 L , {\displaystyle \ {\tfrac {2}{3}}\ L\ ,} where L {\displaystyle \ L\ } is the length of the string. In fact, each n {\displaystyle \ n} th characteristic mode, for n {\displaystyle \ n\ } not a multiple of 3, will not have nodes at these points. These other characteristic modes will be vibrating at the positions 1 3 L {\displaystyle \ {\tfrac {1}{3}}\ L\ } and 2 3 L . {\displaystyle \ {\tfrac {2}{3}}\ L~.} If the player gently touches one of these positions, then these other characteristic modes will be suppressed. The tonal harmonics from these other characteristic modes will then also be suppressed. Consequently, the tonal harmonics from the n {\displaystyle \ n} th characteristic characteristic modes, where n {\displaystyle \ n\ } is a multiple of 3, will be made relatively more prominent. In music, harmonics are used on string instruments and wind instruments as a way of producing sound on the instrument, particularly to play higher notes and, with strings, obtain notes that have a unique sound quality or "tone colour". On strings, bowed harmonics have a "glassy", pure tone. On stringed instruments, harmonics are played by touching (but not fully pressing down the string) at an exact point on the string while sounding the string (plucking, bowing, etc.); this allows the harmonic to sound, a pitch which is always higher than the fundamental frequency of the string. == Terminology == Harmonics may be called "overtones", "partials", or "upper partials", and in some music contexts, the terms "harmonic", "overtone" and "partial" are used fairly interchangeably. But more precisely, the term "harmonic" includes all pitches in a harmonic series (including the fundamental frequency) while the term "overtone" only includes pitches above the fundamental. == Characteristics == A whizzing, whistling tonal character, distinguishes all the harmonics both natural and artificial from the firmly stopped intervals; therefore their application in connection with the latter must always be carefully considered. Most acoustic instruments emit complex tones containing many individual partials (component simple tones or sinusoidal waves), but the untrained human ear typically does not perceive those partials as separate phenomena. Rather, a musical note is perceived as one sound, the quality or timbre of that sound being a result of the relative strengths of the individual partials. Many acoustic oscillators, such as the human voice or a bowed violin string, produce complex tones that are more or less periodic, and thus are composed of partials that are nearly matched to the integer multiples of fundamental frequency and therefore resemble the ideal harmonics and are called "harmonic partials" or simply "harmonics" for convenience (although it's not strictly accurate to call a partial a harmonic, the first being actual and the second being theoretical). Oscillators that produce harmonic partials behave somewhat like one-dimensional resonators, and are often long and thin, such as a guitar string or a column of air open at both ends (as with the metallic modern orchestral transverse flute). Wind instruments whose air column is open at only one end, such as trumpets and clarinets, also produce partials resembling harmonics. However they only produce partials matching the odd harmonics—at least in theory. In practical use, no real acoustic instrument behaves as perfectly as the simplified physical models predict; for example, instruments made of non-linearly elastic wood, instead of metal, or strung with gut instead of brass or steel strings, tend to have not-quite-integer partials. Partials whose frequencies are not integer multiples of the fundamental are referred to as inharmonic partials. Some acoustic instruments emit a mix of harmonic and inharmonic partials but still produce an effect on the ear of having a definite fundamental pitch, such as pianos, strings plucked pizzicato, vibraphones, marimbas, and certain pure-sounding bells or chimes. Antique singing bowls are known for producing multiple harmonic partials or multiphonics. Other oscillators, such as cymbals, drum heads, and most percussion instruments, naturally produce an abundance of inharmonic partials and do not imply any particular pitch, and therefore cannot be used melodically or harmonically in the same way other instruments can. Building on of Sethares (2004), dynamic tonality introduces the notion of pseudo-harmonic partials, in which the frequency of each partial is aligned to match the pitch of a corresponding note in a pseudo-just tuning, thereby maximizing the consonance of that pseudo-harmonic timbre with notes of that pseudo-just tuning. == Partials, overtones, and harmonics == An overtone is any partial higher than the lowest partial in a compound tone. The relative strengths and frequency relationships of the component partials determine the timbre of an instrument. The similarity between the terms overtone and partial sometimes leads to their being loosely used interchangeably in a musical context, but they are counted differently, leading to some possible confusion. In the special case of instrumental timbres whose component partials closely match a harmonic series (such as with most strings and winds) rather than being inharmonic partials (such as with most pitched percussion instruments), it is also convenient to call the component partials "harmonics", but not strictly correct, because harmonics are numbered the same even when missing, while partials and overtones are only counted when present. This chart demonstrates how the three types of names (partial, overtone, and harmonic) are counted (assuming that the harmonics are present): In many musical instruments, it is possible to play the upper harmonics without the fundamental note being present. In a simple case (e.g., recorder) this has the effect of making the note go up in pitch by an octave, but in more complex cases many other pitch variations are obtained. In some cases it also changes the timbre of the note. This is part of the normal method of obtaining higher notes in wind instruments, where it is called overblowing. The extended technique of playing multiphonics also produces harmonics. On string instruments it is possible to produce very pure sounding notes, called harmonics or flageolets by string players, which have an eerie quality, as well as being high in pitch. Harmonics may be used to check at a unison the tuning of strings that are not tuned to the unison. For example, lightly fingering the node found halfway down the highest string of a cello produces the same pitch as lightly fingering the node 1 / 3 of the way down the second highest string. For the human voice see Overtone singing, which uses harmonics. While it is true that electronically produced periodic tones (e.g. square waves or other non-sinusoidal waves) have "harmonics" that are whole number multiples of the fundamental frequency, practical instruments do not all have this characteristic. For example, higher "harmonics" of piano notes are not true harmonics but are "overtones" and can be very sharp, i.e. a higher frequency than given by a pure harmonic series. This is especially true of instruments other than strings, brass, or woodwinds. Examples of these "other" instruments are xylophones, drums, bells, chimes, etc.; not all of their overtone frequencies make a simple whole number ratio with the fundamental frequency. (The fundamental frequency is the reciprocal of the longest time period of the collection of vibrations in some single periodic phenomenon.) == On stringed instruments == Harmonics may be singly produced [on stringed instruments] (1) by varying the point of contact with the bow, or (2) by slightly pressing the string at the nodes, or divisions of its aliquot parts ( 1 2 {\displaystyle {\tfrac {1}{2}}} , 1
The Cancer Imaging Archive
The Cancer Imaging Archive (TCIA) is an open-access database of medical images for cancer research. The site is funded by the National Cancer Institute's (NCI) Cancer Imaging Program, and the contract is operated by the University of Arkansas for Medical Sciences. Data within the archive is organized into collections which typically share a common cancer type and/or anatomical site. The majority of the data consists of CT, MRI, and nuclear medicine (e.g. PET) images stored in DICOM format, but many other types of supporting data are also provided or linked to, in order to enhance research utility. All data are de-identified in order to comply with the Health Insurance Portability and Accountability Act and National Institutes of Health data sharing policies. TCIA resources are intended to support: Development of computer aided diagnosis methods (quantitative imaging) Evaluation of unbiased science reproducibility by acceptable standard statistical methods Research on correlation of clinical diagnostic medical images with digital microscopic histological images Exploratory biomarker research for which imaging is a key element Collaboration between cross-disciplinary investigators where imaging is crucial to research on tumor heterogeneity, between patients and within the tumor; tissue temporal response tracking - objective measurements of tumor progression; imaging genomics and Big Data linkages and analysis (clinical, histo-pathology, genomics) TCIA is recognized as a recommended repository for the Scientific Data, PLOS One, and F1000Research journals. It is also listed in the Registry of Research Data Repositories. == History == Prior to the creation of TCIA, the NCI funded development of the National Biomedical Imaging Archive. NBIA is an open-source Web application which was designed to allow the storage and query of DICOM images. TCIA was subsequently initiated in December 2010 to expand data sharing activities by funding a service component which would help address the technical and policy challenges associated with medical imaging research. TCIA leverages open-source tools such as NBIA and Clinical Trials Processor in order to provide its services. == Organization of the archive == The site content is organized into five categories: About Us - Provides a general overview of the site the organizations responsible for operating it. Share Your Data - Provides an overview of how to apply to upload data to the archive. Access the Archive - Provides information about the available data, methods for accessing that data and system usage metrics. Research Activities - Provides information about major research initiatives being conducted using TCIA data as well as information about publication guidelines. Help - Provides information about how to get support using the archive as well as documentation and data usage policies. == Methods for accessing data == Most collections on the Cancer Imaging Archive can be accessed without an account, but a few are restricted to specific users and therefore require an account to access them. TCIA has several ways to browse, filter, and download data. They include: Downloading the entire contents of a collection in bulk Leveraging the NBIA application to filter or search within or across collections Utilizing the RESTful Application programming interface to filter or search within or across collections === Browsing, bulk downloading and access to supporting data === The home page includes a list of all available collections. Basic information about the data such as the cancer type, cancer location, modalities, and number of subjects are also provided. Clicking on a collection name presents a page which describes the data including its original research purpose, how the data were generated, and how it might be useful to other TCIA users. For example, doi:10.7937/K9/TCIA.2015.L4FRET6Z describes the NSCLC-Radiomics-Genomics Collection. In the lower section of the page there are links to search or download the images and any available supporting data in the Data Access tab. Additional tabs provide information about data versions and how to cite the data if used in publications. Many collections contain additional data types such as genomics, patient demographics, treatment details, and expert analyses of the images. This data is usually only found by browsing the collection pages as opposed to searching in NBIA or using the API. === Filtering or searching with NBIA === On each Collection page and also in the main menu of the site there are links to "Search TCIA". This will load the NBIA application which allows simple, advanced and free text searches. Search results follow the conventional DICOM hierarchy of patient -> study -> series. TCIA provides comprehensive documentation on the various features of the NBIA software. === RESTful API === A number of search and download commands are also available through the API. New iterations on the API are released as new versions, so that existing applications developed against older versions of the API continue to function. == Research activities == A list of known publications based on TCIA data is maintained as a convenience to researchers who might want to investigate how it has been used previously. In addition to peer-reviewed publications there are also several major research initiatives described in the Research Activities section of the site. === The CIP TCGA Radiology Initiative for Radiogenomics Research === A large number of collections contain subjects which were analyzed as part of the NIH/NHGRI database known as The Cancer Genome Atlas (TCGA). This offers researchers the ability to correlate clinical images using shared unique identifiers each study that has in TCGA extensive genomic analysis, digital pathology slides and bulk download of individual demographic data and clinical data. A multi-institutional network of investigators volunteering their time is using the data to develop methods to determine prognosis or predict the response to therapy. TCGA collections are designated by nomenclature shared by the TCGA Data Portal (e.g.: TCGA-BRCA, TCGA-GBM, etc). They are subject to a special publication policy which is unique from the other public data on TCIA. === Challenge competitions === TCIA also provides specific data sets used for "Challenge" competitions such as international digital image-focused professional societies like MICCAI, SPIE, or ISBI. A directory of previous and upcoming challenges is maintained on the site. === Digital object identifiers === To facilitate data sharing, many publications encourage authors to include data citations to the data that the authors used in creating the results described in their scholarly papers. In addition, new journals are now available for describing data collections outright (e.g., Nature Scientific Data). TCIA assigns digital object identifiers (DOIs) to all collections when they are submitted, and also has the ability to create persistent identifiers linked to subsets of data held within TCIA that authors may use for data citations in their scholarly papers.
Fuzzy number
A fuzzy number is a generalization of a regular real number in the sense that it does not refer to one single value but rather to a connected set of possible values, where each possible value has its own weight between 0 and 1. This weight is called the membership function. A fuzzy number is thus a special case of a convex, normalized fuzzy set of the real line. Just like fuzzy logic is an extension of Boolean logic (which uses absolute truth and falsehood only, and nothing in between), fuzzy numbers are an extension of real numbers. Calculations with fuzzy numbers allow the incorporation of uncertainty on parameters, properties, geometry, initial conditions, etc. The arithmetic calculations on fuzzy numbers are implemented using fuzzy arithmetic operations, which can be done by two different approaches: (1) interval arithmetic approach; and (2) the extension principle approach. A fuzzy number is equal to a fuzzy interval. The degree of fuzziness is determined by the a-cut which is also called the fuzzy spread.
Document AI
Document AI, also known as Document Intelligence, refers to a field of technology that employs machine learning (ML) techniques, such as natural language processing (NLP). These techniques are used to develop computer models capable of analyzing documents in a manner akin to human review. Through NLP, computer systems are able to understand relationships and contextual nuances in document contents, which facilitates the extraction of information and insights. Additionally, this technology enables the categorization and organization of the documents themselves. The applications of Document AI extend to processing and parsing a variety of semi-structured documents, such as forms, tables, receipts, invoices, tax forms, contracts, loan agreements, and financial reports. == Key features == Machine learning is utilized in Document AI to extract information from both printed and digital documents. This technology recognizes images, text, and characters in various languages, aiding in the extraction of insights from unstructured documents. The use of this technology can improve the speed and quality of decision-making in document analysis. Additionally, the automation of data extraction and validation can contribute to increased efficiency in document analysis processes. Since the early 2020s, the integration of large language models has extended Document AI beyond extraction toward generative tasks, including the automated drafting of forms, contracts, and document summaries. == Example == A business letter contains information in the form of text, as well as other types of information, such as the position of the text. For instance, a typical letter contains two addresses before the body of the text. The address at the very top (sometimes aligned to the right) is the sender address. This is normally followed by the date of the letter, with the place of writing. After this, the receiver address is listed. The distinction between the sender address and the receiver address is conveyed solely by the position of the address on the page, i.e. there is no textual indication like Sender: in front of the addresses. == Data dimensions and ML architecture == Data is typically distinguished into spatial data and time-series data, the former includes things like images, maps and graphs, while the latter includes signals such as stock prices or voice recordings. Document AI combines text data, which has a time dimension, with other types of data, such as the position of an address in a business letter, which is spatial. Historically in machine learning spatial data was analyzed using a convolutional neural network, and temporal data using a recurrent neural network. With the advent of dimension-type agnostic transformer architecture, these two different types of dimension can be more easily combined, Document AI is an example of this. == Benchmarks == Several public datasets are used to evaluate Document AI systems. FUNSD (Form Understanding in Noisy Scanned Documents) contains 199 annotated forms with token- and block-level labels for form understanding tasks. CORD (Consolidated Receipt Dataset) supports key information extraction from receipts. DocVQA contains approximately 50,000 questions over 12,000 document images for layout-aware visual question answering. == Common uses == Document AI systems are used to automate document processing and information extraction in business and financial workflows, including invoice and receipt processing, data entry automation, anomaly detection, mortgage processing, loan portfolio monitoring, credit risk management, and fraud detection such as counterfeit currency and fraudulent checks. They are also applied in regulatory compliance and contract analysis, including assessing changes in legal and regulatory documents. In real estate, Document AI supports document classification and structured information extraction for standardized processing and analytics. With the adoption of generative AI, Document AI systems can also generate and pre-fill structured documents such as contracts or business forms from natural language prompts.
Content-based image retrieval
Content-based image retrieval, also known as query by image content (QBIC) and content-based visual information retrieval (CBVIR), is the application of computer vision techniques to the image retrieval problem, that is, the problem of searching for digital images in large databases (see this survey for a scientific overview of the CBIR field). Content-based image retrieval is opposed to traditional concept-based approaches (see Concept-based image indexing). "Content-based" means that the search analyzes the contents of the image rather than the metadata such as keywords, tags, or descriptions associated with the image. The term "content" in this context might refer to colors, shapes, textures, or any other information that can be derived from the image itself. CBIR is desirable because searches that rely purely on metadata are dependent on annotation quality and completeness. == Comparison with metadata searching == An image meta search requires humans to have manually annotated images by entering keywords or metadata in a large database, which can be time-consuming and may not capture the keywords desired to describe the image. The evaluation of the effectiveness of keyword image search is subjective and has not been well-defined. In the same regard, CBIR systems have similar challenges in defining success. "Keywords also limit the scope of queries to the set of predetermined criteria." and, "having been set up" are less reliable than using the content itself. == History == The term "content-based image retrieval" seems to have originated in 1992 when it was used by Japanese Electrotechnical Laboratory engineer Toshikazu Kato to describe experiments into automatic retrieval of images from a database, based on the colors and shapes present. Since then, the term has been used to describe the process of retrieving desired images from a large collection on the basis of syntactical image features. The techniques, tools, and algorithms that are used originate from fields such as statistics, pattern recognition, signal processing, and computer vision. === QBIC - Query By Image Content === The earliest commercial CBIR system was developed by IBM and was called QBIC (Query By Image Content). Recent network- and graph-based approaches have presented a simple and attractive alternative to existing methods. While the storing of multiple images as part of a single entity preceded the term BLOB (Binary Large OBject), the ability to fully search by content, rather than by description, had to await IBM's QBIC. === VisualRank === == Technical progress == The interest in CBIR has grown because of the limitations inherent in metadata-based systems, as well as the large range of possible uses for efficient image retrieval. Textual information about images can be easily searched using existing technology, but this requires humans to manually describe each image in the database. This can be impractical for very large databases or for images that are generated automatically, e.g. those from surveillance cameras. It is also possible to miss images that use different synonyms in their descriptions. Systems based on categorizing images in semantic classes like "cat" as a subclass of "animal" can avoid the miscategorization problem, but will require more effort by a user to find images that might be "cats", but are only classified as an "animal". Many standards have been developed to categorize images, but all still face scaling and miscategorization issues. Initial CBIR systems were developed to search databases based on image color, texture, and shape properties. After these systems were developed, the need for user-friendly interfaces became apparent. Therefore, efforts in the CBIR field started to include human-centered design that tried to meet the needs of the user performing the search. This typically means inclusion of: query methods that may allow descriptive semantics, queries that may involve user feedback, systems that may include machine learning, and systems that may understand user satisfaction levels. == Techniques == Many CBIR systems have been developed, but as of 2006, the problem of retrieving images on the basis of their pixel content remains largely unsolved. Different query techniques and implementations of CBIR make use of different types of user queries. === Query By Example === QBE (Query By Example) is a query technique that involves providing the CBIR system with an example image that it will then base its search upon. The underlying search algorithms may vary depending on the application, but result images should all share common elements with the provided example. Options for providing example images to the system include: A preexisting image may be supplied by the user or chosen from a random set. The user draws a rough approximation of the image they are looking for, for example with blobs of color or general shapes. This query technique removes the difficulties that can arise when trying to describe images with words. === Semantic retrieval === Semantic retrieval starts with a user making a request like "find pictures of Abraham Lincoln". This type of open-ended task is very difficult for computers to perform - Lincoln may not always be facing the camera or in the same pose. Many CBIR systems therefore generally make use of lower-level features like texture, color, and shape. These features are either used in combination with interfaces that allow easier input of the criteria or with databases that have already been trained to match features (such as faces, fingerprints, or shape matching). However, in general, image retrieval requires human feedback in order to identify higher-level concepts. === Relevance feedback (human interaction) === Combining CBIR search techniques available with the wide range of potential users and their intent can be a difficult task. An aspect of making CBIR successful relies entirely on the ability to understand the user intent. CBIR systems can make use of relevance feedback, where the user progressively refines the search results by marking images in the results as "relevant", "not relevant", or "neutral" to the search query, then repeating the search with the new information. Examples of this type of interface have been developed. === Iterative/machine learning === Machine learning and application of iterative techniques are becoming more common in CBIR. === Other query methods === Other query methods include browsing for example images, navigating customized/hierarchical categories, querying by image region (rather than the entire image), querying by multiple example images, querying by visual sketch, querying by direct specification of image features, and multimodal queries (e.g. combining touch, voice, etc.) == Content comparison using image distance measures == The most common method for comparing two images in content-based image retrieval (typically an example image and an image from the database) is using an image distance measure. An image distance measure compares the similarity of two images in various dimensions such as color, texture, shape, and others. For example, a distance of 0 signifies an exact match with the query, with respect to the dimensions that were considered. As one may intuitively gather, a value greater than 0 indicates various degrees of similarities between the images. Search results then can be sorted based on their distance to the queried image. Many measures of image distance (Similarity Models) have been developed. === Color === Computing distance measures based on color similarity is achieved by computing a color histogram for each image that identifies the proportion of pixels within an image holding specific values. Examining images based on the colors they contain is one of the most widely used techniques because it can be completed without regard to image size or orientation. However, research has also attempted to segment color proportion by region and by spatial relationship among several color regions. === Texture === Texture measures look for visual patterns in images and how they are spatially defined. Textures are represented by texels which are then placed into a number of sets, depending on how many textures are detected in the image. These sets not only define the texture, but also where in the image the texture is located. Texture is a difficult concept to represent. The identification of specific textures in an image is achieved primarily by modeling texture as a two-dimensional gray level variation. The relative brightness of pairs of pixels is computed such that degree of contrast, regularity, coarseness and directionality may be estimated. The problem is in identifying patterns of co-pixel variation and associating them with particular classes of textures such as silky, or rough. Other methods of classifying textures include: Co-occurrence matrix Laws texture energy Wavelet transform Orthogonal transforms (discrete Chebyshev moments) =
Integreat
Integreat (former project name: Refguide+) is an open source mobile app that provides local information and services tailored to refugees and migrants coming to Germany. The content is maintained by local organizations, such as local governments or integration officers, and made available in locally relevant languages. It was developed by Tür an Tür - Digitalfabrik gGmbH (formerly Tür an Tür - Digital Factory gGmbH) in Augsburg together with a team of researchers and students from the Technical University of Munich. == History == In 1997, the Augsburg association "Tür an Tür", which has been working for refugees since 1992, published the brochure "First Steps", which answers local everyday questions. Since addresses and contact persons change quickly, some information is already outdated after a few weeks. Students of business informatics at the Technical University of Munich therefore developed the app Integreat within eight months together with the association and the social department of the city of Augsburg. The app was then also used by other cities and districts within months. As of February 3, 2022, information is available at 72 locations, including Munich, Dortmund, Nuremberg and Augsburg. == Mode of action == Refugees need information on areas such as registration, contact persons, health care, education, family, work and everyday life. Integreat seeks to provide refugees with this information by allowing them to select their geographic location and receive locally relevant information. This information is available offline once the app is opened so it can be used without an internet connection. In addition, the content is translated into the native languages of refugees and migrants to facilitate access. The content is licensed with a CC BY 4.0 license to facilitate collaboration and translation between content creators and dissemination of the content. Integreat is now being used for a broader migrant audience and says it can also support professionals, volunteers, and counseling centers. == Comparable mobile apps == Other mobile apps that are likewise intended to provide initial orientation for refugees include the app Ankommen, a joint project of the Federal Office for Migration and Refugees, the Goethe-Institut, the Federal Employment Agency and the Bavarian Broadcasting Corporation, which is intended as a companion for the first few weeks in Germany, and the Welcome App, a company-sponsored non-profit initiative for information about Germany and asylum procedures with a regional focus, and a book by the Konrad Adenauer Foundation (KAS) and Verlag Herder with a corresponding app Deutschland - Erste Informationen für Flüchtlinge (Germany - First Information for Refugees) as a companion for Arabic-speaking refugees in Germany.
Fuzzy differential inclusion
Fuzzy differential inclusion is the extension of differential inclusion to fuzzy sets introduced by Lotfi A. Zadeh. x ′ ( t ) ∈ [ f ( t , x ( t ) ) ] α {\displaystyle x'(t)\in [f(t,x(t))]^{\alpha }} with x ( 0 ) ∈ [ x 0 ] α {\displaystyle x(0)\in [x_{0}]^{\alpha }} Suppose f ( t , x ( t ) ) {\displaystyle f(t,x(t))} is a fuzzy valued continuous function on Euclidean space. Then it is the collection of all normal, upper semi-continuous, convex, compactly supported fuzzy subsets of R n {\displaystyle \mathbb {R} ^{n}} . == Second order differential == The second order differential is x ″ ( t ) ∈ [ k x ] α {\displaystyle x''(t)\in [kx]^{\alpha }} where k ∈ [ K ] α {\displaystyle k\in [K]^{\alpha }} , K {\displaystyle K} is trapezoidal fuzzy number ( − 1 , − 1 / 2 , 0 , 1 / 2 ) {\displaystyle (-1,-1/2,0,1/2)} , and x 0 {\displaystyle x_{0}} is a trianglular fuzzy number (-1,0,1). == Applications == Fuzzy differential inclusion (FDI) has applications in Cybernetics Artificial intelligence, Neural network, Medical imaging Robotics Atmospheric dispersion modeling Weather forecasting Cyclone Pattern recognition Population biology